1. Dirac’s Quantum Condition
Classical mechanics relates two conjugated variables by using the Poisson bracket. Dirac’s quantum condition extends this relation to quantum mechanical operators:
The commutator between two operators must relate to the classical Poisson bracket between the two corresponding functions through the following relationship
HW #2
Operators which follow Dirac’s quantum condition form an internally consistent set,
though there may be more than one set (different representations)

2. Poisson bracket and commutator
For example, for f(x,p)= x and g(x,p)=p the Poisson bracket is

3. Variance
Finally, we can connect everything we know about commutators and the Dirac’s quantum condition and obtain the most fundamental property of the Quantum World
individual measurements
expectation value

4. Uncertainty Principle
For the product of the standard deviations of two properties of a quantum mechanical system whose state wavefunction is y, it can be shown (you’ll do it in your HW2)

5. The Born Interpretation
The explicit function representing a state of a system in a particular coordinate system and in a particular representation is called WAVEFUNCTION: 
Remember? Probability of an event is given by |f|2 where f is a complex number (probability amplitude)

6. cont
Normalization: The probability of finding the particle SOMEWHERE in space and time MUST be =1
Wavefunctions which obey the eigenvalue equation are called eigenfunctions